Estimation and compensation of frequency sweep nonlinearity in FMCW radar

Estimation and compensation of

frequency sweep nonlinearity in FMCW radar

by Kurt Peek

Picture: www.thalesgroup.com

A thesis submitted in partial fulfillment Of the requirements for the degree of MASTER OF SCIENCE in APPLIED MATHEMATICS

at

The University of Twente

Under the supervision of dr. ir. G. Meinsma and prof. dr. A.A. Stoorvogel

September 2011

Table of Contents

1 Introduction ... 4

1.1 ‘Hardware’ sweep linearization ... 7

1.2 ‘Software’ linearization techniques ... 8

1.3 This thesis ... 9

2 Theory of operation of FMCW radar ... 10

2.1 Analytical model of a FMCW radar ... 10

2.2 The effect of sinusoidal nonlinearity in the frequency sweep ... 18

3 An algorithm for compensating the effect of phase errors on the FMCW beat signal spectrum 23 3.1 Prior work ... 23

3.2 Mathematical preliminaries ... 24

3.3 Description of the phase error compensation algorithm ... 25

3.4 Derivation of the algorithm for temporally infinite chirps ... 28

3.5 Application of the algorithm to finite chirps ... 34

4 Simulation ... 43

4.1 Digital implementation of the phase error compensation method... 43

4.2 Implementation of the deskew filter by the frequency sampling method ... 43

4.3 Simulation of the phase error compensation algorithm ... 50

4.4 Results for cases of interest ... 53

4.5 Concluding remarks... 57

5 Estimation of the phase errors ... 59

5.1 Review of a known method using a reference delay ... 59

5.2 Proposal of a novel method using ambiguity functions ... 59

6 Conclusions and discussion ... 61

Bibliography ... 62

Appendix: MATLAB simulation ... 66

Abstract

One of the main issues limiting the range resolution of linear frequency-modulated continuous-wave (FMCW) radars is nonlinearity of frequency sweep, which results in degradation of contrast and range resolution, especially at long ranges. Two novel, slightly different, methods to correct for nonlinearities in the frequency sweep by digital post-processing of the deramped signal were introduced independently by Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003) and Meta et al.

(Meta, Hoogeboom et al. 2006). In these publications, however, no formal proof of the techniques

was given, and no limitations were described. In this thesis, we prove that the algorithm of Meta is

exact for temporally infinite chirps, and remains valid for finite chirps with large time-bandwidth

products provided the maximum frequency component of the phase error function is sufficiently

low. It is also shown that the algorithm of Meta reduces to that of Burgos-Garcia in this limit. A

digital implementation of both methods described. We also propose a novel method to measure the

systematic phase errors which are required as input to the compensation algorithm.

1 Introduction

Frequency-modulated continuous-wave (FMCW) radars provide high range measurement precision and high range resolution at moderate hardware expense (Griffiths 1990; Stove 1992). Moreover, the spreading of the transmitted power over a large bandwidth provides makes FMCW radar difficult to detect by intercept receivers, providing it with stealth in military applications. In the last two decades, Thales Netherlands has developed a family of silent radars for air surveillance, coastal surveillance, navigation, and ground surveillance based on FMCW technology.

In FMCW radar, the range to the target is measured by systematically varying the frequency of a transmitted radio frequency (RF) signal. Typically, the transmitted frequency is made to vary linearly with time; for example, a sawtooth or triangular frequency sweep is implemented. The linear variation of frequency with time is often referred to as a chirp, frequency sweep, or frequency ramp, and is associated with a quadratically increasing phase (see Section 2.1.1). Figure 1 shows a time- frequency plot of a linear sawtooth FMCW transmit signal and its corresponding amplitude.

Figure 1 (a) Time-frequency plot of a FMCW transmit signal with carrier frequency 𝒇𝒄, sweep period 𝑻, and bandwidth¹ 𝑩. Typical parameters are 𝒇𝒄 = 10 GHz, 𝑻 = 500 μs, and 𝑩 = 50 MHz. (b) Time-amplitude plot of a transmitted FMCW signal (not with the parameters listed above).

1

The term ‘bandwidth’ is often used in this context to refer to the total excursion of the instantaneous frequency during one the sweep period. The FMCW signal is not bandlimited in the mathematical sense of the word; however, for large time-bandwidth products it is approximately bandlimited.

time

time instantaneous

frequency

amplitude

bandwidth 𝐵 = 50 MHz

sweep period 𝑇 = 500 µs carrier

frequency 𝑓

= 10 GHz

(a)

(b)

The frequency sweep effectively places a “time stamp” on the transmitted signal at every instant, and the frequency difference between the transmitted signal and the signal returned from the target (i.e. the reflected or received signal) can be used to provide a measurement of the target range, as illustrated in Figure 2. This process is called dechirping or deramping, and the frequency of the dechirped signal is called the beat or intermediate frequency (IF) signal.

Figure 2 Principle of FMCW range measurement. (a) Time-frequency plots of the transmitted chirp (solid line) and the echoes from two ‘point’ targets (dashed lines), delayed by their respective two-way propagation delays to the target and back. (b) Time-frequency plots of the frequency difference, or ‘beat frequency’, between the transmitted and received chirps. The beat frequency is observed during portion of the sweep period in which the transmitted and received signals overlap.

As seen from Figure 2, the transit time to the target and back and the target beat frequency are directly proportional, and their proportionality constant is equal to the chirp rate (i.e., the ratio between the bandwidth and the sweep period) of the transmitted chirp. Hence, the target transit time – and thus, the target range – can be inferred by a measurement of the beat frequency.

The beat frequency is generated in the receiver of the FMCW radar by a mixer

or ‘multiplier’ as illustrated in Figure 3. The local oscillator (LO) port of the mixer is fed by a portion of the transmit

2

A mixer is a three-port device that uses a nonlinear or time-varying element to achieve frequency conversion (Pozar 2005). In its down-conversion configuration, it has two inputs, the radio frequency (RF) signal and the

the product of RF and LO signals.

transmitted linear chirp

received target echoes

frequency after dechirping instantaneous

transmit frequency

time

time target beat

signals target round-

trip delays

target beat

frequencies

signal

, while the radio frequency (RF) port is fed by the target echo signal from the receive antenna.

As explained in more detail in Section 2.1.3, the output of the mixer, called the intermediate frequency (IF) signal, has a phase which (after low-pass filtering) is equal to the difference of the phases of the LO and RF input signals. Hence, its frequency is the ‘beat’ frequency described above.

The beat signal is passed to a spectrum analyzer, which is a bank of filters used to resolve the target returns into range bins. Typically, the spectrum analyzer is implemented as an analog-to-digital converter (ADC) followed by a processor based on the fast Fourier transform (FFT).

Figure 3 Simplified block diagram of a homodyne FMCW radar transceiver. A chirp generator generates a linear sawtooth FMCW signal (left, upper inset) which is radiated out to the target scene by a transmit antenna. A portion of the transmitted signal is coupled to the local oscillator (LO) port of a mixer. The target echo received by a separate receive antenna is fed to the radio frequency (RF) port of the mixer. The mixer output at intermediate frequency (IF) is fed to a spectrum analyzer. The output of the spectrum analyzer for a single target is a ‘sinc’ function centered at the target beat frequency (left, lower inset).

The performance of linear FMCW radar depends critically on the linearity of the transmitted signal.

Deviation of the instantaneous frequency of a FMCW chirp from linearity – or, equivalently, deviation of its phase from a quadratic – causes ‘smearing’ of the target beat signal in frequency, resulting in the appearance of spurious sidelobes or “ghost” targets and degradation of the signal-to- noise ratio (SNR). The effect is usually worse at larger range, where phases of the transmitted and received signals are more de-correlated. This effect is illustrated in Figure 4.

3

This is the homodyne receiver architecture, in which the local oscillator signal is provided by the transmitter itself. Alternatively, the local oscillator can be generated separately and triggered at an appropriate instant;

this is commonly referred to as stretch processing (Caputi 1971). Stretch processing has the disadvantage of the additional complexity of another oscillator. Receiver noise effects will also be greater because of the independence of the phase noise of the separate oscillators (Piper 1993).

chirp generator

spectrum analyzer time

coupler

mixer

transmit antenna

freq u enc y

receive antenna

target RF

LO

IF

frequency

power

Figure 4 FMCW range measurement with non-linear chirps. Due to the nonlinearity of the transmitted chirp, the target beat signals are ‘spread’ or ‘smeared’ in frequency. The degradation worsens with increasing range.

A number of different approaches have thus been adopted over the years with the aim of improving the frequency sweep linearity of FMCW radar systems. These can be categorized in ‘hardware’

techniques, which attempt to generate highly linear chirps, and ‘software’ techniques, which use signal processing to compensate the effects of the non-linearity a posteriori. Although our focus in this report is on the latter, it is instructive to discuss shortly the former.

1.1 ‘Hardware’ sweep linearization

Firstly, attempts have been made to produce chirp generators that are inherently linear. One way is to apply a linear sawtooth current signal to a Yttrium Iron Garnet (YIG)-tuned oscillator, which is a current-controlled oscillator (CCO) with an inherently linear tuning characteristic. This scheme is representative of the world’s first mass-produced FMCW navigation radar: the Pilot FMCW radar, developed by Philips Research Laboratories in 1988 and marketed by its subsidiaries PEAB in Sweden and Hollandse Signaalapparaten in the Netherlands (Pace 2009). The typically attainable sweep linearity of 0.1% still limits the obtainable range resolution in FMCW applications, however, and the switching speed is low, of the order of hundreds of microseconds. Finally, with this technique the phase varies slightly from sweep to sweep; this limits the performance of signal processing methods which are based on the coherent operation of the FMCW radar, such as Doppler processing (Barrick 1973) and coherent integration (Beasley 2009).

In FMCW transmitters employing voltage-controlled oscillators (VCOs), the most common

‘hardware’ method used for frequency sweep linearization is closed loop feedback. The closed loop transmitted

non-linear chirp

received target echoes

frequency after dechirping instantaneous

transmit frequency

time

time target beat

signals

feedback technique has been implemented in a variety of ways, but they are all based on creating an artificial target which generates a “beat” frequency when mixed with a reference signal. In a

perfectly linearized FMCW radar a fixed range target would produce a constant “beat” frequency.

Therefore, in a practical FMCW radar, if the “beat” frequency drifts from the desired constant frequency value, an error signal can be generated to fine-tune the VCO to maintain a constant

“beat” frequency. This feedback technique can be implemented at the final RF frequency of the radar or at a lower, down-converted frequency. Waveforms having sweep linearity

better than 0.05% have been demonstrated (Fuchs, Ward et al. 1996) but, unless the system is very well designed, the technique can be prone to instabilities and is typically limited in bandwidth to about 600 MHz. Also, because the VCO is modulated directly, the phase noise of the resultant transmit signal can be compromised (Beasley 2009). Finally, the use of sweep linearization precludes

coherent operation of the radar, because the feedback loop “does its own thing” during each sweep period, so that the phase in each sweep is independent from that in the other sweeps.

The use of a direct digital synthesizer (DDS) offers quite a cost-effective solution, however the transmitted bandwidth is still limited compared to the one obtained by directly sweeping a VCO.

Moreover, nonlinearity can still be caused by group delay in subsequent filters (Perez-Martinez, Burgos-Garcia et al. 2001).

Method References Advantages Disadvantages

Free-running YIG oscillator

PILOT FMCW radar (Beasley, Leonard et al. 2010)

Low phase noise, sweep linearity of 0.1% attainable

Slow modulation speed, power-hungry, drifts with temperature, phase varies slightly from sweep to sweep

VCO with closed-loop feedback

(Fuchs, Ward et al.

1996)

Sweep linearity better than 0.05%

Prone to instabilities, typically limited in bandwidth to about 600 MHz, precludes coherent operation

Direct digital synthesis (DDS)

(Goldberg 2006) Linear chirp generated to digital precision

Limited bandwidth, requires upconversion, group delay introduced by filtering further down the transmission chain

Table 1 Techniques for generating linear FMCW chirps.

Table 1 summarizes techniques for generating linear chirps with their advantages and disadvantages.

In short, each of the ‘hardware’ techniques has its limitations.

1.2 ‘Software’ linearization techniques

As an interesting alternative to these hardware techniques, a software-based linearization method using a transmission measurement through a reference delay line has been reported in both FMCW radar (Fuchs, Ward et al. 1996; Vossiek, Kerssenbrock et al. 1997) and, more recently, in optical frequency-domain reflectometers (OFDR) (Ahn, Lee et al. 2005; Saperstein, Alic et al. 2007). These methods involve resampling the beat signal at so-called “constant phase intervals” so that it signal

4

Sweep linearity is defined as the maximum deviation in frequency from a linear chirp as a percentage of the

swept bandwidth.

has linear behavior. The resampling can be achieved both by hardware or software (Nalezinski, Vossiek et al. 1997). A drawback of this technique, however, it that it assumes that the phase error can be linearized on the target delay interval, which limits its validity to short range intervals (Meta, Hoogeboom et al. 2007).

Relatively recently, Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003) and Meta et al. (Meta, Hoogeboom et al. 2007) have reported on novel processing methods which employ a “residual video phase” (RVP) or “deskew” filter. These methods, which operate directly on the deramped data, correct the nonlinearity effects for the whole range profile at once, and are based only on the assumption that the transmitted chirp has a large time-bandwidth product.

1.3 This thesis

The algorithms proposed by Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003) and Meta et al.

(Meta, Hoogeboom et al. 2007) are actually slightly different. Further, they are presented based on heuristic reasoning; no formal proof is given, and no limitations of the algorithm are mentioned.

This thesis makes three main contributions to knowledge:

(1) We give a proof of both Meta’s algorithm, which is valid for wideband IF signals, and Burgos- Garcia’s algorithm, which is valid for narrowband IF signals. It is shown that the algorithm of Meta reduces to that of Burgos-Garcia in the special case that the error frequency

components are sufficiently low. Further, our analytical results indicate that the original algorithm as presented by Meta (Meta, Hoogeboom et al. 2006; Meta, Hoogeboom et al.

2007) contains a sign error. Finally, we discuss issues which arise when applying the algorithm to time-limited chirps, which have not been discussed previously.

(2) We implement both phase error compensation algorithms in MATLAB and demonstrate their effectiveness. (In (Burgos-Garcia, Castillo et al. 2003) and (Meta, Hoogeboom et al.

2007), no detail was given on the digital implementation of the algorithm). The results of our simulation are inconclusive, however, as to whether there is a sign error in Meta’s derivation or not. Further improvements to the simulation algorithm, which involve taking into account the “edge effects” due to the time-limited nature of the chirps, are proposed.

(3) We propose a novel method for determining the phase errors using measurements from targets at several different reference delays, based on the synthesis problem of a function from its ambiguity function as discussed by Wilcox (Wilcox 1991). The novel method could have advantages over known methods, which use just a single reference delay.

The organization of this thesis is as follows. In Chapter 2, we discuss the theory of operation of

FMCW radar in mathematical detail, and review how phase errors affect their operation. In Chapter

3, we derive both Meta’s and Burgos-Garcia’s variations of the phase error compensation algorithm

analytically, and address the issues mentioned above in point (1). In Chapter 3, we perform a

simulation of the algorithms and demonstrate their effectiveness. In Chapter 5, we discuss the

estimation of phase errors, which are required as input for the algorithm. Finally, in Chapter 6, we

wrap up with our conclusions and discussion.

2 Theory of operation of FMCW radar

This chapter presents a tutorial review of the basic principles of FMCW (frequency modulated continuous wave) radars. The material to follow is on homodyne FMCW radar, i.e., CW radar in which a microwave oscillator is frequency-modulated and serves as both transmitter and local oscillator (Skolnik 2008). The effect of frequency sweep nonlinearity is also discussed.

An outline of this chapter is as follows. In Section 2.1, we present an analytical model of the generation of the target range profile by a FMCW transmitting ideal linear sawtooth chirps. In Section 2.2, we discuss how its performance is affected by sinusoidal frequency sweep nonlinearities.

2.1 Analytical model of a FMCW radar

In this section, we explain the principle FMCW range measurement in more mathematical detail. In Section 2.1.1, we formulate an expression for the transmitted signal. In Section 2.1.2, we construct a model for the received signal. In Section 2.1.3, we explain the generation of the ‘dechirped’,

‘deramped’, or ‘beat’ signal. Of particular importance for the algorithm to be described is the use of

‘coherent detection’ to obtain complex samples of this signal. Finally, in Section 2.1.4, we discuss the spectrum of the beat signal or ‘video signal’, which is used to visualize the target scene.

2.1.1 Transmitted signal

We select a 100% duty factor signal whose frequency sweeps upward, linearly, over one sweep repetition interval 𝑇. Using a complex number representation (Jakowatz, Wahl et al. 1996), the transmitted signal 𝑠

with unity amplitude can be expressed as the real part of

𝑠

𝑡 = 𝑠

𝑡 − 𝑛𝑇

∞ 𝑛=−∞

, (2.1)

where 𝑠

𝑡 is the linear chirp pulse 𝑠

𝑡 = rect 𝑡

𝑇 exp 𝑗2𝜋 𝑓

𝑡 + 1

2 𝛼𝑡

≡ rect 𝑡

𝑇 exp 𝑗𝜙

𝑡 . (2.2) Here 𝑡 represents the time variable, 𝑗 = −1 the imaginary unit, 𝑓

the chirp’s center frequency, and 𝛼 its frequency sweep rate, and rect ∙ is the rectangular function given by

rect 𝑥 =

1, 𝑥 < 1 2 , 1

2 , 𝑥 = 1 2 , 0, 𝑥 > 1

2 .

(2.3)

We assume here that the transmit signal is periodic, and hence phase-coherent from one sweep to the next

.

5

By sweep-to-sweep coherence, we mean that there is a fixed relationship between the phase in one sweep

and the next, i.e., 𝜙

𝑡 + 𝑇 − 𝜙

𝑡 = constant. FMCW radars having this property are called coherent,

and have several advantages. For example, coherent systems allow Doppler processing (Barrick 1973) to

determine information on the velocity of detected targets. Furthermore, coherent integration over 𝑁

Since the instantaneous frequency, 𝑓

𝑡 , is the derivative of the phase (Carson 1922), we have 𝑓

𝑡 = 1

2𝜋 𝑑𝜙

𝑑𝑡 = 𝑓

+ 𝛼𝑡. (2.4)

Thus it can be seen that the frequency excursion over one sweep repetition interval is 𝛼𝑇 = 𝐵, the chirp bandwidth. The instantaneous frequency of the transmit signal is plotted in Figure 5(a) as the solid line.

Figure 5 Time-frequency plots of (a) the transmitted (solid line) and received (dashed line) signals, and (b) the intermediate frequency (IF) signal. The IF alternates between two distinct tones: 𝒇𝒃𝟏= 𝜶𝝉 for intervals of duration 𝑻 − 𝝉 and 𝒇𝒃𝟐= −𝜶 𝑻 − 𝝉 for intervals of duration 𝝉, where 𝜶 = 𝑩/𝑻 is the frequency sweep rate. Typical chirp parameters for an FMCW navigation radar are 𝒇𝒄 = 10 GHz, 𝑩 = 50 MHz, and 𝑻 = 500 μs.

2.1.2 Received signal

After transmission of the radar signal through the transmit antenna, the radar waveform propagates to the target scene, and part of the energy is scattered back to the radar’s receive antenna. In the following analytical development, we assume that the target scene consists of a single stationary

‘point’ target such that the echo signal 𝑠

𝑡 is simply a delayed replica of the transmit signal:

𝑠

𝑡 = 𝑠

𝑡 − 𝜏 , (2.5)

where 𝜏 is the two-way propagation delay given by

frequency sweeps improves the signal-to-noise ratio (SNR) by a factor of 𝑁. This should be contrasted with the SNR increase of 𝑁 typically obtained using incoherent integration of 𝑁 frequency sweeps (Beasley 2009).

𝜏

time

𝑓

= 𝛼𝜏 𝑇

𝐵 𝑓

𝑇 − 𝜏

𝑓

= −𝛼 𝑇 − 𝜏 (a)

(b)

time target echo

transmitted chirp

beat signal frequency

frequency

0

𝜏 = 2𝑅

𝑐 , (2.6)

where 𝑅 is the range of the stationary ‘point’ target, and 𝑐 is the propagation velocity.

If we assume that the radar receiver is a linear system

, then the range profile obtained from a general target scene can be obtained by superposition of the range profiles of the individual targets.

Thus, the modeling a ‘point’ target is merely a convenient way to separate algorithm and hardware effects from target and interference phenomenology.

To obtain an expression for the received signal corresponding to a single sweep of the transmitted signal, we insert (2.2) into (2.5) to find

𝑠

𝑡 = rect 𝑡 − 𝜏

𝑇 exp 𝑗2𝜋 𝑓

𝑡 − 𝜏 + 1

2 𝛼 𝑡 − 𝜏

≡ rect 𝑡 − 𝜏

𝑇 exp 𝑗𝜙

𝑡 . (2.7) The instantaneous frequency of the periodic repetition of 𝑠

, 𝑠

, is plotted in Figure 5(a) as the dashed line.

2.1.3 Dechirped signal

As explained in the Introduction, upon reception the received signal is correlated with the transmitted signal through a mixing process. In this section, we explain in more detail the mixing process and subsequent digitization (Section 2.1.3.1) and the retrieval of phase information by a method called in-phase (𝐼) / quadrature (𝑄) demodulation (Section 2.1.3.2).

2.1.3.1 Mixing process

Now after bandpass filtering to reject wideband noise and radio frequency (RF) amplification, the received signal is ‘dechirped’ or ‘deramped’ by ‘mixing’ or ‘beating’ it together with a replica of the transmitted signal in a mixer as illustrated in Figure 6. The resulting signal will contain a product term 𝐺 cos 𝜙

cos 𝜙

, where 𝐺 is a constant accounting for the voltage conversion loss of the mixer, and other higher-order products. In general, only the lowest-order product will have significant amplitude. The product may be expanded as a sum, namely

𝐺

2 cos 𝜙

− 𝜙

+ cos 𝜙

+ 𝜙

.

The phase-sum term represents an oscillation at twice the carrier frequency, which is generally filtered out either actively, or more usually in radar systems because it is beyond the cut-off frequency of the mixer and subsequent receiver components (Brooker 2005)

. We thus obtain the

6

In practice, the FMCW receiver is not an ideally linear system; for example, nonlinear behavior of the mixer and high-gain pre-amplifier which follows the receive antenna causes harmonic distortion and intermodulation distortion (IMD). These are separate hardware issues however, however; here, we are concerned with errors arising from nonlinearity of the frequency sweep.

7

FMCW radars sometimes employ a so-called image reject mixer (IRM) instead of a conventional one to

generate the IF signal. The FMCW radar using a conventional mixer suffers a 3 dB loss in signal-to-noise ratio

(SNR) due to the addition of noise at the RF image frequency to the RF noise when both are down-converted

to near-zero IF. This effect cannot easily be removed by RF filtering because of the closeness of the RF and

image frequencies, but can be removed if an IRM is used (Willis and Griffiths 2007).

function

cos 𝜙

− 𝜙

, which is called the ‘dechirped’, ‘deramped’, ‘beat’, or ‘intermediate frequency’ (IF) signal. The IF signal with unity amplitude (we do not consider amplitude variations in this derivation) is thus

cos 𝜙

≡ cos 𝜙

− 𝜙

. (2.8)

As shown in Figure 6, the IF signal is sampled in by an analog-to-digital (A/D) converter after low- pass filtering to prevent wideband noise from folding into range of interest of target beat frequencies.

Figure 6 Simplified block diagram of a homodyne FMCW transceiver. The received (RX) signal is fed to the radio frequency (RF) port of a mixer, while a portion of the transmit (TX) signal is coupled to the local oscillator (LO) port. The mixer output is low-pass filtered to obtain the desired intermediate frequency (IF) signal, which is digitized by an analog- to-digital (A/D) converter at a rate 𝒇𝒔 of at least twice the maximum beat frequency 𝒇𝒃,𝒎𝒂𝒙.

A complex representation of the IF signal resulting from a single pulse of the transmitted signal is obtained by Inserting (2.2) and (2.7) into (2.8), a single pulse of the IF signal can be expressed as the real part of

𝑠

𝑡 ≡ 𝑠

𝑡 𝑠

𝑡

= rect 𝑡

𝑇 rect 𝑡 − 𝜏

𝑇 exp 𝑗2𝜋 𝑓

𝑡 + 1

2 𝛼𝑡

− 𝑓

𝑡 − 𝜏 − 1

2 𝛼 𝑡 − 𝜏

or, simplifying,

𝑠

𝑡 = 𝑟 𝑡 exp 𝑗2𝜋 𝑓

𝜏 + 𝛼𝜏𝑡 − 1

2 𝛼𝜏

, (2.9)

where the beat signal envelope 𝑟 𝑡 is given by

𝑟 𝑡 = 1, −𝑇/2 + 𝜏 < 𝑡 < 𝑇/2,

0, otherwise. (2.10)

During the remaining part of the sweep period, on the interval − 𝑇 2 , − 𝑇 2 + 𝜏 , the received signal corresponds to the transmitted signal during the previous sweep. Therefore, the mixer output will be offset by the sweep width, 𝐵, as illustrated in Figure 5(b). 𝐵 is much greater than the signal frequency and the mixer output for − 𝑇 2 < 𝑡 < − 𝑇 2 + 𝜏 will therefore be filtered and rejected.

Hence, for − 𝑇 2 < 𝑡 < − 𝑇 2 + 𝜏, the IF signal will be a transient pulse. If a digital data system is used to observe the mixer output, the sampling can be delayed at the start of each sweep so the retrace effects of the local oscillator returning to 𝑓

− 𝐵 2 are simply omitted (Strauch 1976).

A/D mixer low-pass

filter RF

LO

TX out RX in

from transmitter directional

coupler

IF

𝑓

≥ 2𝑓

Let us consider the three terms that comprise the phase of the IF signal (2.9):

 𝑓

𝜏 is the total number of cycles of 𝑓

that occur during the round trip propagation time for the target

.

 𝛼𝜏𝑡 is a term increasing linearly with the time 𝑡, and represents the target beat frequency 𝑓

= 𝛼𝜏.

 −𝛼𝜏

/2 is a range-dependent phase term. In the synthetic aperture radar (SAR) literature, it is called the residual video phase (Carrara, Goodman et al. 1995). As we will see, this term plays a key role in the phase compensation algorithm.

The third term, the residual video phase, will prove to play a crucial role in the phase error compensation algorithm.

2.1.3.2 Retrieval of in-phase and quadrature components The mixing process described above produces a real voltage signal

ℛℯ 𝑠

= cos 𝜙

, (2.11)

where 𝑠

is given by (2.9). After analog-to-digital conversion as described in Section 2.1.4, this appears digitally as an array of real numbers. Ideally, however, we would like to obtain the complex representation 𝑠

itself, which we refer to as the baseband signal. Knowledge of the baseband signal has a number of advantages:

 It allows positive and negative frequencies to be recovered separately. As pointed out by Gurgel and Schlick (Gurgel and Schlick 2009), in the case of a linear chirp with increasing frequency (a positive chirp), the beat frequency defined by (2.9) will always be positive.

Therefore, a 3 dB gain in signal-to-noise ratio (SNR) can be obtained by avoiding the aliasing noise from the “unused” negative side of the spectrum.

 By converting the IF signal into baseband form, a simple multiplication of each sample with the appropriate complex number achieves any desired phase adjustment of that sample.

The latter point is an essential part of the phase compensation algorithm to be described in the following chapter. Thus, it is desirable to obtain the beat signal in complex form, but how is this done?

Mathematically, there are actually two ways of obtaining a complex representation of a signal from a real one (Boashash 1992):

1) The “real plus imaginary quadrature” representation, in which the cosine in (2.11) is

replaced by a complex exponential. The real and imaginary parts of this complex exponential are called the in-phase (𝐼) and quadrature (Q) components, respectively:

𝐼 = ℛℯ 𝑠

= cos 𝜙

, 𝑄 = ℐ𝓂 𝑠

= sin 𝜙

. (2.12)

8

Incidentally, for coherent FMCW radar applications such as Doppler processing and coherent integration, this

term should ideally be constant for all processed sweeps. However, because this term typically is very large

compared to the other terms, this places very stringent requirements on the frequency stability of the chirp

generator (Strauch 1976).

2) The analytic signal representation, in which the negative frequency components of (2.11) are discarded and the positive ones multiplied by a factor two. (This is equivalent to adding to (2.11) an imaginary part equal to its Hilbert transform).

As shown by Nuttall (Nuttall and Bedrosian 1966), the two approaches are only equivalent if the

“real plus imaginary quadrature” representation is spectrally one-sided. In our case, it is “real plus imaginary quadrature” representation that corresponds exactly with the desired baseband signal.

The conversion of real signals to a baseband representation, 𝐼 + 𝑗𝑄, is performed by a so-called I/Q demodulator, also known as a quadrature detector, synchronous detector, or coherent detector (Skolnik 2008). Coherent detection can be performed both before and after digitization.

Figure 7 illustrates the classical analog implementation of an I/Q demodulator. The received signal cos 𝜙

is split and fed to a pair of mixers or analog multipliers. The transmit signal cos 𝜙

, obtained from the transmit chain by a directional coupler, is input to a quadrature splitter, also known as a quadrature hybrid or 90° hybrid (Pozar 2005). Ideally, this results in two outputs: one proportional to cos 𝜙

in phase with the input, and the other proportional to sin 𝜙

at phase quadrature to the input. These outputs are fed to LO ports of two mixers and mixed (multiplied) with the received signal, cos 𝜙

. As in Section 2.1.3.1, the mixer products can be expanded into phase- sum and phase-difference terms via the trigonometric relations

cos 𝜙

cos 𝜙

= 1

2 cos 𝜙

+ 𝜙

+ cos 𝜙

− 𝜙

and

cos 𝜙

sin 𝜙

= 1

2 sin 𝜙

+ 𝜙

+ sin 𝜙

− 𝜙

.

The sum-frequency components are at approximately twice the RF frequency and easily filtered.

What remains are the terms cos 𝜙

− 𝜙

= cos 𝜙

and sin 𝜙

− 𝜙

= sin 𝜙

. These

are exactly the in-phase (𝐼) and quadrature (𝑄) components of the IF signal, respectively, and can be

combined to obtain the full baseband signal 𝑠

= exp 𝑗𝜙

as desired.

Figure 7 Simplified block diagram of an analog I/Q demodulator for a homodyne FMCW system. The received signal is applied to a 3-dB power splitter, the two outputs of which are applied to the RF ports of (double-balanced) mixers. The local oscillator (LO) ports are driven by two samples of the transmit signal, the two components being in phase quadrature. The resulting outputs from the mixers are low-pass filtered and digitized by analog-to-digital (A/D) converters to obtain the in-phase (𝑰) and quadrature (𝑸) components representative of the received vector.

Although the classical analog I/Q demodulator provides a clear example of how baseband conversion can be implemented, in most modern systems I/Q demodulation is performed after digitization. This has the advantage of avoiding so-called “I/Q mismatch” problems which hamper the analog implementation (Pun, Franca et al. 2003). The flipside of this, however, is that digital I/Q demodulators require a rate that is twice that of each of the A/D converters in Figure 7; in effect, complex sampling requires real sampling at twice the rate. Given the high sample rates obtainable with modern A/D converters, however, this is usually not a problem.

There are actually several techniques, referred to as “direct sampling digital coherent detection techniques” (Pun, Franca et al. 2003), for performing baseband conversion after digitization, which were developed in the early 1980s (Rice and Wu 1982; Waters and Jarrett 1982; Rader 1984). A detailed discussion of these techniques is beyond the scope of this thesis; we will simply use the result that the signal 𝑠

given by (2.9) can be obtained in baseband form.

2.1.4 Video signal

After quadrature sampling of the IF signal 𝑠

, a processor based on the fast Fourier transform (FFT) resolves the beat frequency spectrum into frequency and range bins. Following Stove (Stove 1992), we refer to the beat signal after frequency analysis as the video signal.

As explained in Section 2.1.3, the portion of the IF signal that is within the receiver bandwidth is a pulse train with pulse length 𝑇 − 𝜏 and pulse repetition interval 𝑇. Since the IF signal is periodic with period 𝑇, its target range information can be obtained from the Fourier transform 𝑆

of a single pulse:

splitter

quadrature splitter

A/D

A/D

TX out RX in

from transmitter

in-phase component 𝐼

3 dB

directional coupler

quadrature component 𝑄 0°

90°

mixer low-pass filter RF

LO

RF LO

𝑓

≥ 2𝑓

𝑆

𝑓 = 𝑠

𝑡 exp −𝑗2𝜋𝑓𝑡 𝑑𝑡

−∞

, (2.13)

where 𝑠

𝑡 is given by (2.9). Here and throughout this thesis, functions represented by uppercase letters are Fourier transforms of the functions represented by the corresponding lowercase letters.

Substituting (2.9) into (2.13), evaluating the Fourier integral, and taking its absolute value, we find 𝑆

𝑓 = 𝑇 − 𝜏 sinc 𝑇 − 𝜏 𝑓 − 𝛼𝜏 , (2.14) where sinc ∙ is the normalized “sinc” function defined as

sinc 𝑥 ≡ sin 𝜋𝑥 𝜋𝑥 .

An illustrative plot of the amplitude spectrum is shown in Figure 8. The target at range 𝑅 = 𝑐𝜏 2 shows up as a peak at the target beat frequency 𝑓

= 𝛼𝜏. Although the spectrum is plotted as a function of frequency, the abscissa can be scaled by a factor 𝑐𝑇 2𝐵 to yield a plot of target reflectivity versus range; this plot is called the range profile.

Figure 8 Amplitude spectrum 𝑺𝑰𝑭 𝒇 of one pulse of the in-band portion of the IF signal. The spectrum as is a “sinc”

shaped peak at the target beat frequency 𝒇𝒃= 𝜶𝝉, where 𝜶 is the sweep rate and 𝝉 the two-way propagation delay. The width 𝚫𝒇𝒃 of the peak (strictly, width at -3.9 dB) is the reciprocal of the duration of the pulse, 𝟏 𝑻 − 𝝉 .

As stated in the Introduction, in practice the spectral analysis is performed by an analog-to-digital converter (ADC) followed by a processor based on the fast Fourier transform (FFT). Approximating the spectrum in this way involves a number of practical considerations:

 Nyquist criterion. Typically, the FMCW radar is only designed to detect targets up to a certain maximum range or instrumented range 𝑅

. In order to prevent aliasing of the spectra of targets within 𝑅

, the ADC sample rate 𝑓

should be chosen at least twice the maximum beat frequency 𝑓

:

Δ𝑓

= 1 𝑇 − 𝜏

frequency 𝑓 amplitude

spectrum 𝑆

𝑓

𝑓

= 𝛼𝜏

𝑓

≥ 2𝑓

. (2.15) In order to prevent out-of-band noise from folding back into the target spectrum, the beat signal is typically passed through an anti-aliasing filter, which is a low-pass filter with a cutoff frequency between the maximum beat frequency 𝑓

and the Nyquist frequency 𝑓

/2.

 ADC interval. As explained in Section 2.1.3, during the initial 𝜏

seconds of each sweep, a portion of the beat signal for targets within the instrumented range is outside the bandwidth of the ADC. This interval is usually omitted by delaying the sampling by 𝜏

seconds from the beginning of each sweep, or alternatively by setting the samples collected during the initial 𝜏

to zero (Adamski, Kulpa et al. 2000). As a result, the spectral width of a ‘point’

target is Δ𝑓

= 1 𝑇 − 𝜏

for all targets within the instrumented range.

 Sidelobe apodization. The beat signal spectrum 𝑆

𝑓 given by (2.14) has the characteristic

‘sinc’ shape as predicted by Fourier theory. The range side lobes in this case are only 13.3 dB lower than the main lobe, which is not satisfactory as it can result in the occlusion of small nearby targets as well as introducing clutter from the adjacent lobes into the main lobes. To counter these undesirable effects, a window function (Harris 1978) is usually applied to the sampled IF signal prior to FFT frequency estimation. In our simulation in Chapter 3, we employ a Hamming window with a highest sidelobe level of -43 dB.

These practical aspects are of importance in explaining our simulation in Chapter 3

. To summarize, we have analyzed the generation and spectral analysis of the beat signal in

mathematical detail for ideal, linear frequency sweeps. In the following section, we investigate how this process is affected if the sweeps are nonlinear – in particular, if they are perturbed by sinusoidal frequency sweep nonlinearity.

2.2 The effect of sinusoidal nonlinearity in the frequency sweep

This section presents an analysis describing the effects on the range resolution of homodyne linear FMCW radar of sinusoidal nonlinearities in the frequency sweep. Our discourse follows the analyses of Richter (Richter, Jensen et al. 1973), Griffiths (Griffiths 1991; Griffiths and Bradford 1992), and Piper (Piper 1995).

2.2.1 Analytical development

We treat the chirp signal as a nominally-linear FM sweep of rate 𝛼 and unit amplitude, with the frequency error expressed in terms of a departure from frequency sweep linearity with amplitude 𝛿

and frequency 𝑓

:

𝑓

𝑡 = 𝑓

+ 𝛼𝑡 + 𝛿

cos 2𝜋𝑓

𝑡 , − 𝑇

2 < 𝑡 < 𝑇

2 . (2.16)

This non-linear time-frequency characteristic is illustrated together with its linear counterpart in Figure 9.

9

As explained in 2.1.3.2, another important digital signal processing step is the digital I/Q demodulation. In our

simulation in Chapter 3Se, we assume this has already been done, and used complex samples directly.

Figure 9 Time-frequency characteristics of a linear chirp (blue line) and a non-linear chirp (red curve). The linear chirp on the interval −𝑻/𝟐, 𝑻/𝟐 has a center frequency 𝒇𝒄, duration 𝑻, chirp rate 𝜶, and frequency deviation (or ‘bandwidth’) 𝑩 = 𝜶𝑻. The non-linear chirp is

The phase of the transmitted signal 𝜙

is obtained by integrating the instantaneous angular frequency 𝜔

= 2𝜋𝑓

in accordance with (2.4). Arbitrarily setting 𝜙

= 0 at 𝑡 = 0 (there is no loss of generality here), we thus have the relation

𝜙

𝑡 = 2𝜋 𝑓

𝑡

0

𝑑𝑡

. (2.17)

Inserting the expression (2.16) for 𝑓

into (2.17), we find 𝜙

𝑡 = 2𝜋 𝑓

𝑡 + 1

2 𝛼𝑡

+ 𝐴

sin 2𝜋𝑓

𝑡 , (2.18) where 𝐴

= 𝛿

/𝑓

is the “modulation index” of the transmitted chirp, i.e., its maximum phase error.

The phase of the beat signal is given by (cf. (2.8))

𝜙

𝑡 = 𝜙

𝑡 − 𝜙

𝑡 − 𝜏 , (2.19)

where 𝜏 is the target transit time as defined by (2.6). Inserting (2.18) into (2.19) yields 𝜙

𝑡 = 2𝜋 𝑓

𝜏 + 𝛼𝜏𝑡 − 1

2 𝛼𝜏

+ 𝐴

sin 2𝜋𝑓

𝑡 − sin 2𝜋𝑓

𝑡 − 𝜏 (2.20) or, using trigonometric identities to factorize the IF phase error term,

𝜙

𝑡 = 2𝜋 𝑓

𝜏 + 𝛼𝜏𝑡 − 1

2 𝛼𝜏

+ 2𝐴

sin 𝜋𝑓

𝜏 cos 2𝜋𝑓

𝑡 − 𝜏

2 . (2.21) The baseband dechirped signal with envelope 𝑟 𝑡 is therefore given by (cf. (2.9)):

𝑡 𝑓

1/𝑓

𝛿

𝑓

−𝑇/2 𝑇/2

𝛼𝑇 linear chirp

non-linear chirp

𝑠

𝑡 = 𝑟 𝑡 exp 𝑗𝜙

𝑡

= 𝑟 𝑡 exp 𝑗 𝜙

+ 2𝜋𝛼𝜏𝑡 + 𝛽 cos 2𝜋𝑓

𝑡 − 𝜏

2 (2.22)

where 𝜙

= 2𝜋 𝑓

𝜏 − 𝛼𝜏

is a constant phase term and 2

𝛽 ≡ 2𝐴

sin 𝜋𝑓

𝜏 (2.23)

is the “modulation index”, or maximum phase error, of the IF signal.

The expression (2.22) is recognizable from narrowband phase modulation theory. It can be expanded as

𝑠

𝑡 = 𝑟 𝑡 exp 𝑗 𝜙

+ 2𝜋𝛼𝜏𝑡 1 + 𝑗𝛽 cos 2𝜋𝑓

𝑡 − 𝜏 2

− 1

2! 𝛽

cos

2𝜋𝑓

𝑡 − 𝜏

2 + ⋯ (2.24)

Now, if we assume that the peak phase error is small, i.e.,

𝛽 ≪ 1, (2.25)

then only the first two terms of the expansion in (2.24) are significant. Thus the baseband dechirped signal is approximately

𝑠

𝑡 ≈ 𝑟 𝑡 exp 𝑗 𝜙

+ 2𝜋𝛼𝜏𝑡 1 + 𝛽

2 exp 𝑗2𝜋𝑓

𝑡 − 𝜏 2 + 𝛽

2 exp −𝑗2𝜋𝑓

𝑡 − 𝜏

2 (2.26) which is the distortionless point-target response, plus a pair of sidelobes, or paired echoes, at ±𝑓

. The amplitude of each of these sidebands is 𝛽/2.

2.2.1.1 Limit of long-wavelength phase errors

For long-wavelength phase errors such that the sidelobe ripple period is much larger than the target transit time, i.e.,

1

𝑓

≫ 𝜏, (2.27)

(2.23) is well approximated by

𝛽 ≈ 𝐴

𝜔

𝜏, (2.28)

where 𝜔

≡ 2𝜋𝑓

is the angular ripple frequency. Thus, for long-wavelength phase errors, the modulation parameter 𝛽 in the beat signal increases linearly with the target transit time 𝜏, and hence with target range. Physically, we can say that for delays which are small compared to the wavelength of the phase error, the transmitted and received phase errors ‘cancel out’

. 2.2.1.2 General phase errors

In the preceding discussion, we considered a sinusoidal phase error. Here, we argue that the above analysis can be extended to general phase errors.

10

This is in contrast to conventional pulse compression radars, in which the ‘paired echo’ effect is independent

of target range (Klauder 1960). As a result, requirements on frequency sweep linearity can be considerably less

stringent for FMCW radar than for LFM pulse compression radar, as noted by Griffiths (Griffiths 1991).

A general phase error can be written in the form

𝑓

𝑡 = 𝑓

+ 𝛼𝑡 + 𝑓

𝑡 , − 𝑇

2 < 𝑡 < 𝑇

2 . (2.29)

The error frequency 𝑓

𝑡 can be expanded as a Fourier series:

𝑓

𝑡 = 𝑎

2 + 𝑎

cos 2𝜋𝑛𝑡

𝑇 + 𝑏

sin 2𝜋𝑛𝑡 𝑇

∞ 𝑛 =1

. (2.30)

where

𝑎

= 1

𝜋 𝑓

2𝜋𝑡

𝑇 cos 𝑛𝑥 𝑑𝑥

𝜋

−𝜋

, 𝑏

= 1

𝜋 𝑓

2𝜋𝑡

𝑇 sin 𝑛𝑥 𝑑𝑥

𝜋

−𝜋

. (2.31)

Now, the constant term 𝑎

/2 in (2.30) has only the effect of changing the center frequency 𝑓

𝜏 of the chirp, and since the center frequency is present in the beat signal phase only in the constant phase term 𝑓

𝜏, this term has no effect on the amplitude spectrum of the beat signal.

Thus, neglecting the constant frequency term 𝑎

/2 and integrating (2.30), we find that the phase error 𝜙

𝑡 is given by

𝜙

𝑡 = 2𝜋 𝑓

𝑡

0

𝑑𝑡

= 𝑎

𝑇

𝑛 sin 2𝜋𝑛

𝑇 𝑡 − 𝑏

𝑇

𝑛 cos 2𝜋𝑛 𝑇 𝑡

∞ 𝑛=1

.

(2.32)

(Here we have omitted a constant phase term which also has no effect on the range profile).

The phase error in the IF signal, 𝜙

, is the difference between the transmitted phase error 𝜙

and its version delayed by 𝜏:

𝜙

𝑡 = 𝜙

𝑡 − 𝜙

𝑡 − 𝜏 . (2.33) Inserting (2.32) into (2.33), subtracting term by term, and applying trigonometric identities as

before, we find

𝜙

𝑡 = 2 sin 𝜋𝑛

𝑇 𝜏 𝑎

𝑇

𝑛 cos 2𝜋𝑛 𝑇 𝑡 − 𝜏

2 + 𝑏

𝑇

𝑛 sin 2𝜋𝑛 𝑇 𝑡 − 𝜏

2

∞ 𝑛=1

. (2.34)

Now, a little thought shows that if we substitute (2.34) for the single-tone phase error

𝛽 cos 2𝜋𝑓

𝑡 − 𝜏/2 in (2.22) and expand the factor exp 𝑗𝜙

as a Taylor series, then the higher-order terms can be neglected as long as 𝜙

is small compared to unity. A sufficient condition for this is that the amplitudes of the frequency errors are much smaller than the sweep repetition frequency, i.e.,

𝑎

≪ 1

𝑇 and 𝑏

≪ 1

𝑇 . (2.35)

In this case, the target beat spectrum consists of a superposition of ‘paired echoes’ spaced at multiples of the sweep repetition frequency, 1/𝑇, from the desired target beat signal.

In short, within small-angle approximations for the phase error, the ‘paired echoes’ associated with the harmonics of the phase error merely superpose. Hence, an algorithm that compensates the

‘paired echoes’ for a chirp perturbed by sinusoidal phase errors and is linear should also work for

general phase errors, provided that these errors are sufficiently small. The derivation of such phase

error compensation algorithm is the subject of the next chapter.

3 An algorithm for compensating the effect of phase errors on the FMCW beat signal spectrum

In this chapter, we present a novel algorithm for compensating the effect of phase errors on the FMCW beat signal spectrum by digital post-processing of the beat signal. Given amount of effort that is currently put into making chirps linear, the existence of this algorithm is a very significant in the field of FMCW ranging, and could render such elaborate chirp linearization methods obsolete.

This chapter is organized in three sections. In Section 3.1, we discuss similar algorithms that were devised by others, and highlight the differences with our approach. In Section 3.2, we establish some mathematical preliminaries – namely the quadratic phase filter and the Fresnel transform – which will allow us to describe the algorithm more succinctly. In Section 3.3, we present a flow chart describing the algorithm. In Section 3.4, we present an analytical derivation of the algorithm for temporally infinite chirps, and show that the algorithm is exact in this case. In Section 3.5, we apply the algorithm to finite chirps, and show that it remains approximately valid for chirps with large time-bandwidth product and for which the phase error function contains only low frequencies.

3.1 Prior work

A signal processing method was devised, apparently independently, by Burgos-Garcia et al. (Burgos- Garcia, Castillo et al. 2003) and Meta et al. (Meta, Hoogeboom et al. 2006; Meta, Hoogeboom et al.

2007) to compensate for non-linearities in the frequency sweep (or equivalently, phase errors in the phase) of FMCW signals. (Actually, the system described in (Burgos-Garcia, Castillo et al. 2003) is a heterodyne time-domain pulse compression radar instead of a homodyne FMCW radar, but the results can be applied to the latter case). The algorithm, which operates directly on deramped data, corrects non-linearity effects for the whole range at once, and is computationally efficient.

Burgos-Garcia et al. and Meta et al. present the algorithm in a slightly different form, which is also different from the one described here. In particular (as we will explain in more detail in Section 3.4),

1. In the last step of the algorithm described by Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003), the phase error function in the receive signal, which they call 𝜙

𝑡 , is used directly to cancel the residual phase error after removal of the transmitted errors and range deskew. This is based on their stated assumption that the beat signal from the 𝑖th target is a narrowband signal centered at the frequency 𝑓

= 𝛼𝜏

. Our derivation shows that this assumption is not necessary, and that a skew-filtered version of the phase error function can be used in the case that the beat signal is not narrowband

.

2. The algorithm described by Meta (Meta, Hoogeboom et al. 2006) does use a filtered version of the phase error function in the last step. However, this version is a Fresnel transform of the phase error function, whereas we believe it should be an inverse Fresnel transform

.

11

The author initiated a private e-mail correspondence with Mr. Burgos-Garcia, but unfortunately he was not at liberty to discuss the details of the algorithm under the terms of his project contract with the Spanish defense company Indra EWS.

12

The author also e-mailed Mr. Meta about this, but unfortunately he was too busy to study the derivation.

Interestingly, an international an international patent application was submitted for this technique (Meta

2007), but at the time of writing is deemed to be withdrawn.

Moreover, (Burgos-Garcia, Castillo et al. 2003) and (Meta, Hoogeboom et al. 2006) use heuristic arguments to justify the steps, and no formal proof of the algorithm was given. Our analytical derivation in Section 3.4 is thus a novel contribution to the literature on this subject.

3.2 Mathematical preliminaries

The key component of the phase error compensation (PEC) algorithm to be described is the

quadratic phase filter (QPF). As a prelude to our presentation of the algorithm, here we first discuss the properties of this filter, as well as an integral transform called the Fresnel transform associated with it (Gori 1994; Papoulis 1994).

3.2.1 The quadratic phase filter

A QPF is an all-pass system with quadratic phase. We denote by 𝑞

𝑡 its impulse response and by 𝑄

𝑓 its transfer function:

𝑞

𝑡 = −𝑗𝛼 exp 𝑗𝜋𝛼𝑡

↔ 𝑄

𝑓 = exp −𝑗𝜋 𝑓

𝛼 . (3.1)

where the double arrow (↔) denotes a Fourier transform pair, and the time and frequency domains are identified by the arguments 𝑡 and 𝑓, respectively. Note that since neither 𝑞

𝑡 nor 𝑄

𝑓 is square-integrable, this Fourier transform pair should be interpreted in the generalized sense as the limit as 𝜎 → 0

of the Fourier transform of the complex Gaussian beam exp −𝜋 𝜎 − 𝑗𝛼 𝑡

, where 𝜎 and 𝛼 are real parameters (Papoulis 1977).

The QPF is a dispersive filter which introduces a group delay proportional to the frequency. Group delay is a measure of the time delay of the amplitude envelope of a sinusoidal component; it is in general different from the phase delay, which is the time delay of the phase. The group delay of a constant-modulus filter 𝐻 𝑓 = exp 𝑗Φ 𝑓 is given by

𝑡

𝑓 = − 1 2𝜋

𝑑Φ

𝑑𝑓 . (3.2)

Thus, the group delay 𝑡

(𝑓) of the QPF given by (3.1) is 𝑡

𝑓 = 𝑓

𝛼 . (3.3)

Hence, the group delay of a QPF is linearly proportional to the frequency. (Incidentally, the group delay of a QPF is the inverse function of the instantaneous frequency of its impulse response:

𝑓

𝑡 = 𝛼𝑡. This result does not hold in general, but holds here because 𝑞

𝑡 is a so-called

‘asymptotic’ signal (Boashash 1992)).

3.2.2 The Fresnel transform

The Fresnel transform with chirp parameter 𝛼 of a function 𝑠 𝑡 , denoted 𝑠

𝑡 here, is by definition the output of a QPF with input 𝑠 𝑡 :

𝑠

𝑡 = −𝑗𝛼 𝑠 𝑡

exp 𝑗𝜋𝛼 𝑡 − 𝑡

𝑑𝑡

−∞

= 𝑠 𝑡 ∗ 𝑞

𝑡 , (3.4)

where the asterisk (∗) denotes the convolution product. The inversion formula reads

𝑠 𝑡 = 𝑗𝛼 𝑠

𝑡 exp −𝑗𝜋𝛼 𝑡 − 𝑡

𝑑𝑡

−∞

= 𝑠

𝑡 ∗ 𝑞

𝑡 , (3.5)

so that the inverse transform simply equals the Fresnel transform with parameter −𝛼.

Let us denote by 𝑆 𝑓 the Fourier transform of a function 𝑠 𝑡 . From the definition (3.4) and the convolution theorem, it follows that the Fourier transform 𝑆

𝑓 of 𝑠

𝑡 equals

𝑆

𝑓 = 𝑆 𝑓 exp −𝑗𝜋 𝑓

𝛼 . (3.6)

It is also useful to investigate an asymptotic limit of the Fresnel transform. Based on the limit (Papoulis 1977)

𝛼→∞

lim −𝑗𝛼 exp 𝑗𝜋𝛼𝑡

= 𝛿 𝑡 , (3.7)

where 𝛿 ∙ denotes the Dirac delta function, it follows that in the limit 𝛼 → ∞ the Fresnel transform of a function approaches the function itself, i.e.,

𝛼→∞

lim 𝑠

𝑡 = 𝑠 𝑡 . (3.8)

The Fresnel transform manifests itself several areas of signal and image processing, including pulse compression, fiber-cable communications and dispersion, and Fresnel diffraction and optical filtering (Papoulis 1994). Here, it will allow us to give a concise description of the PEC algorithm.

3.3 Description of the phase error compensation algorithm

Suppose the transmitted signal is perturbed by a phase error 2𝜋𝜖(𝑡). We assume that 𝜖 𝑡 is known (its estimation is discussed in Chapter 4), and define the phase error function

𝑠

𝑡 ≡ exp 𝑗2𝜋𝜖 𝑡 . (3.9)

The correction algorithm, shown schematically in Figure 10, consists of the following three steps:

1. The complex-valued deramped data 𝑠

𝑡 is first multiplied by the complex conjugate of the phase error function, 𝑠

𝑡 , in order to eliminate phase errors resulting from transmitted non-linearities.

2. The resulting signal, 𝑠

(𝑡), is then passed through a deskew filter

with frequency response

𝑄

𝑓 = exp 𝑗𝜋 𝑓

𝛼 , (3.10)

where 𝛼 is the (nominal) chirp rate of the transmitted chirp. Thus, the deskew filter is a QPF with a parameter a negative group delay −𝑓/𝛼, which has the effect of aligning the received

13

Incidentally, the deskew filter is commonly used in synthetic aperture radar (SAR) signal processing, where

its purpose is to remove the range-dependent phase term −𝜋𝛼𝜏

from the beat signal (cf. Section 2.1.3) of

relevance in SAR applications. Since this phase term is called the residual video phase (RVP), the filter is also

called an ‘RVP filter’ (Carrara, Goodman et al. 1995) in this context.